Nonlinear Control Sy.. - Free

218 CHAPTER 8. PASSIVITY
Definition 8.6 Consider a rational function H(s) = p(s)/q(s), where E P', and
E P'. Then H(s) is said to be in the class Q if
(i) is a Hurwitz polynomial (i.e. the roots of are in the open left half plane), and
(ii) Re[H(jw)] > 0, Vw E [0, oo).
Definition 8.7 H(s), is said to be weak SPR if it is in the class Q and the degree of the
numerator and denominator polynomials differ by at most 1. H(s) is said to be SPR if it
is weak SPR and in addition, one of the following conditions is satisfied:
(i) n = m, that is, p and q have the same degree.
(ii) n = m + 1, that is, H(s) is strictly proper, and
lim w2JIe[H(yw)] > 0
W-i00
It is important to notice the difference amon,, the several concepts introduced above.
Clearly, if H(s) is SPR (or even weak SPR), then H(s) E Q. The converse is, however, not
true. For example, the function Hl (s) = (s + 1)-1 + s3 E Q, but it is not SPR according
to Definition 8.5. In fact, it is not even positive real. The necessity of including condition
(ii) in Definition 8.7 was pointed out by Taylor in Reference [79]. The importance of this
condition will became more clear soon.
We now state an important result, known as the Kalman-Yakubovich lemma.
Lemma 8.1 Consider a system of the form
Lb = Ax+Bu, xEl[t", uE1im
y = Cx+Du, UERm
and assume that (i) the eigenvalues of A lie in the left half of the complex plane, (ii)
(A, B) is controllable, and (iii) (C, A) is observable. Then H = C(sI - A)-1B + D is
SPR if and only if there exist a symmetric positive definite matrix P E Rn,n, and matrices
Q E Rmxm W E lRm', and e > 0 sufficiently small such that
PA + ATP = _QQT _.EP (8.26)
PB + WTQ = C (8.27)
WTW = D+DT (8.28)